dancing with maths chris budd. what have the following got in common?
Post on 20-Dec-2015
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Dancing with maths
Chris Budd
What have the following
got in common?
A snowflake
A starfish
Tilbury Fort
Escher drawing
Folk dancing
They all have symmetry
Symmetry is the basis of all patterns
In art, music, bell ringing, knitting, dancing, crystals, elementary particles and nature
Some types of symmetry
Reflexion
Rotation
Translation
Something is symmetric if it is not changed by one of these operations
Lots of good artistic patterns have this property
A square is very symmetric … how
Many symmetries does it have?
8
4 Rotation symmetries
4 Reflexion symmetries
Rotation
Reflexion
Reflexion
a
b
c
Simplest symmetry .. Do nothing
Call this symmetry e
a rotation of 90 degrees
aa rotation of 180 degrees
aaa rotation of 270 degrees
aaaa rotation of 360 degrees
aaaa =
Can combine symmetries to get new ones
e
bb = e cc = e dd = e ff = e
Can combine reflexions with themselves
What happens if we combine a reflexion with a rotation?
or two different reflexions?
ba = c
Reflexion and rotation = reflexion
Reflexion and rotation = b a = ?
ab = d
So … what is ab
bc = aRemember
This!!!!!
Now combine two reflexions bc = ?
cb = aaa
db = abb = ae = a
Some other combinations
Let’s start dancing!
My name is Chris. I go to a dance with my friends Andrew, Bryony and Daphne
A B C D
We make ABCD four corners of a square
The symmetries of the square correspond to different dance moves
Key Fact
Reflexion
Symmetry:
Dance move:
A B C D A C B D
An inner-twiddle or dos-e-dos
b
b
Reflexion
c
Dance move:
A B C D B A D C
An outer-twiddle or swing
Symmetry:
c
Now for the clever bit!
In the algebra of symmetries
bc = a
Therefore
bc bc bc bc = aaaa = e
Did you remember this?
This corresponds to a dance called a Reel of Four or a Hey
So what?????
Let’s do the dance
ABCD
ACBD
CADB
CDAB
DCBA
DBCA
BDAC
BADC
ABCD
b
c
b
c
b
c
b
c
Now it’s your turn!!
Another dance
d b = a
d b d b d b d b = aaaa = e
ABCD CDABd
ABCD
CDAB
CADB
DBCA
DCBA
BADC
BDAC
ACBD
ABCD
d
b
d
b
d
b
d
b
We see the same patterns in knitting and in bell ringing
And many other places
How many can you find?